# Equivalent Circuit Determination of Quartz Crystals

This application note describes how to measure the equivalent circuit parameters of a quartz crystal with the SARK-110 Antenna Analyzer.

The process for the determination of the crystal parameters is laborious, so in order to simplify this process the SARK-110 provides the automatic determination of the crystal parameters feature. This feature is available in firmware versions v0.8.3.7 or higher.

The Figure 1 shows a screenshot of the measurement of the parameters of a crystal. The model provides the resonant frequencies –series and parallel, the quality factor, and the values of the crystal circuit model.

**Figure 1, Quartz crystal equivalent circuit model screen**

## Equivalent Circuit Measurement

The crystal must be connected to the analyzer by using an appropriate test fixture. In this case it has been used a board edge type SMA connector, soldered to a socket for plug-in the crystal; see Figure 2. The test port adapter is a short SMA to MCX adapter cable.

It is essential that the analyzer is calibrated before each measurement. The open, short and load calibration loads must be connected to the end of the test port extension cable. In addition, it will be convenient performing frequency calibration for the best accuracy on resonance frequency measurements. Please refer to the user’s manual on how to carry out the steps for calibration.

**Figure 2, Crystal test fixture**

The procedure for the determination of the circuit model is the following:

**Figure 3, Equivalent circuit model of an 11 MHz crystal**

The value of the series resonant frequency is transferred to the frequency in Single Frequency mode and to the centre frequency in the sweep modes, e.g. Scalar Chart or Smith Chart modes. This allows graphically examining the characteristics at the resonant frequency; see Figure 4, Figure 5 and Figure 6.

Note that it is important setting the automatic scale for the correct display of the impedance characteristics in the Scalar Chart mode.

**Figure 4, Crystal impedance values at resonant frequency in Single Frequency mode**

**Figure 5, Crystal impedance characteristics at resonant frequency**

**Figure 6, Crystal impedance characteristics at resonant frequency**

The Figure 7 shows the series and parallel resonant frequencies in the Scalar Chart mode. For this crystal it was set a span of 40 kHz for the complete display of those resonant frequencies.

**Figure 7, Series and parallel resonant frequencies**

## How it works

The process starts by automatically looking for the series and parallel resonant frequencies. The start scan frequency is taken from the frequency value in Single Frequency mode. The scan range is +1MHz up and -1MHz down from this value.

The resonant frequencies are identified in the singularities where the impedance changes from pure capacitive (phase value close to -90º) to pure inductive (phase value close to +90º). The resonant frequencies are then obtained from the frequency points where measured phase value is close to zero.

After determining the series and parallel resonant frequencies, it is measured the series resistance (Rs) at the series resonant frequency. Then it is measured the parallel capacitance (Co). This value is measured in a frequency that is 2.5 MHz below Fs or 2.5 MHz above Fp. With all these measurements, it will be derived the rest of parameters:

The value of series capacitance (Cs) is given by:

The value of the series inductance (Ls) is given by:

Finally, the quality factor of the crystal (Q) is calculated by: